Answer:
(a)Charlie is right
(b)$0
Step-by-step explanation:
(a)A game is said to be a fair game when the probability of winning is equal to the probability of losing. Mathematically, a game is said to be fair when the expected value is zero.
In the game, the possible outcomes are: HH, HT, TH and TT.
Charlie wins when the outcome is HH, TT
- P(Charlie Wins)=2/4
- P(Charlie Losses)=2/4
Lucy wins when the outcome is HT or TH
- P(Lucy Wins)=2/4
- P(Lucy Losses)=2/4
Therefore, the game is fair. Charlie is right.
(b)
If the outcome is HH, Lucy pays $3.
If the outcome is HT or TH, Lucy gets $2.
If the outcome is TT, Lucy pays $1.
The probability distribution of Lucy's profit is given below:
![\left|\begin{array}{c|c|c|c}$Profit(x)&-\$3&-\$1&\$2\\P(x)&1/4&1/4&2/4\end{array}\right|](https://tex.z-dn.net/?f=%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7Cc%7Cc%7Cc%7D%24Profit%28x%29%26-%5C%243%26-%5C%241%26%5C%242%5C%5CP%28x%29%261%2F4%261%2F4%262%2F4%5Cend%7Barray%7D%5Cright%7C)
Expected Profit
![=(-3 \times \frac14)+(-1\times \frac14)+(2 \times \frac24)\\=$0](https://tex.z-dn.net/?f=%3D%28-3%20%5Ctimes%20%5Cfrac14%29%2B%28-1%5Ctimes%20%5Cfrac14%29%2B%282%20%5Ctimes%20%5Cfrac24%29%5C%5C%3D%240)
Lucy's expected profit from the game is $0.